Fourier analysis is fundamentally a method for expressing a function as a
sum of periodic components, and for recovering the signal from those
components. When both the function and its Fourier transform are
replaced with discretized counterparts, it is called the discrete Fourier
transform (DFT). The DFT has become a mainstay of numerical computing in
part because of a very fast algorithm for computing it, called the Fast
Fourier Transform (FFT), which was known to Gauss (1805) and was brought
to light in its current form by Cooley and Tukey [CT]. Press et al. [NR]
provide an accessible introduction to Fourier analysis and its
applications.